### Abstract

A matching M in a graph G is said to be extendable if there exists a perfect matching of G containing M. In 1989, it was shown that every connected planar graph with at least 8 vertices has a matching of size three which is not extendable. In contrast, the study of extending certain matchings of size three or more has made progress in the past decade when the given graph is 5-connected planar triangulation or 5-connected plane graphs with few non-triangular faces.In this paper, we prove that if G is a 5-connected plane graph of even order in which at most two faces are not triangular and M is a matching of size four in which the edges lie pairwise distance at least three apart, then M is extendable. A related result concerning perfect matching with proscribed edges is shown as well.

Original language | English |
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Journal | Discrete Mathematics |

DOIs | |

Publication status | Accepted/In press - 2017 |

### Fingerprint

### Keywords

- Distance restricted matching extension
- Plane graph
- Punctured triangulation

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*. https://doi.org/10.1016/j.disc.2017.07.017

**Edge proximity and matching extension in punctured planar triangulations.** / Aldred, R. E.L.; Fujisawa, Jun; Saito, Akira.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Edge proximity and matching extension in punctured planar triangulations

AU - Aldred, R. E.L.

AU - Fujisawa, Jun

AU - Saito, Akira

PY - 2017

Y1 - 2017

N2 - A matching M in a graph G is said to be extendable if there exists a perfect matching of G containing M. In 1989, it was shown that every connected planar graph with at least 8 vertices has a matching of size three which is not extendable. In contrast, the study of extending certain matchings of size three or more has made progress in the past decade when the given graph is 5-connected planar triangulation or 5-connected plane graphs with few non-triangular faces.In this paper, we prove that if G is a 5-connected plane graph of even order in which at most two faces are not triangular and M is a matching of size four in which the edges lie pairwise distance at least three apart, then M is extendable. A related result concerning perfect matching with proscribed edges is shown as well.

AB - A matching M in a graph G is said to be extendable if there exists a perfect matching of G containing M. In 1989, it was shown that every connected planar graph with at least 8 vertices has a matching of size three which is not extendable. In contrast, the study of extending certain matchings of size three or more has made progress in the past decade when the given graph is 5-connected planar triangulation or 5-connected plane graphs with few non-triangular faces.In this paper, we prove that if G is a 5-connected plane graph of even order in which at most two faces are not triangular and M is a matching of size four in which the edges lie pairwise distance at least three apart, then M is extendable. A related result concerning perfect matching with proscribed edges is shown as well.

KW - Distance restricted matching extension

KW - Plane graph

KW - Punctured triangulation

UR - http://www.scopus.com/inward/record.url?scp=85027441616&partnerID=8YFLogxK

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U2 - 10.1016/j.disc.2017.07.017

DO - 10.1016/j.disc.2017.07.017

M3 - Article

AN - SCOPUS:85027441616

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

ER -